My wife received a CR-1, so I can't answer all your questions concerning a K-1 in Taiwan.....but I can give you this information about AIT accepted physicians:
https://www.ait.org.tw/wp-content/uploads/sites/269/2017/06/TAI-Taipei-physicians-06012017-ZHTW.pdf
The actual visa application at the U.S. embassy in Taiwan is just a matter of providing the required documents, completing the medical, etc. .. nothing very complicated. How the embassy in Taiwan processes the visa application can be different than how it is done at the embassy in Manila so you need to follow the process as they have listed on the U.S. embassy website in Taiwan.